Math 6610 • Analysis of Numerical Methods I

{Elliptic Problems}

Consider the one dimensional problem for \(v(x)\)
\[
  v''(x)=f(x), \tag{1}
\]
in the interval \([0,1]\) along with homogeneous Dirichlet boundary conditions.
Define the difference operator
\[
  \Delta_h U_j\equiv\frac{1}{h^2}(U_{j-1}-2U_j+U_{j+1}),
\]
and consider the scheme for Eq. 1
\[
  \Delta_h U_j=f_j
\]
for \(j=1,2,\ldots,N-1\) where \(Nh=1\), \(f_j\equiv f(jh)\), and \(U_0=U_N=0\). The
approximate solution satisfies a linear system \(AU=b\), where
\(U=(U_1,U_2,\ldots,U_{N-1})^T\) and
\(b=h^2(f_1,f_2,\ldots,f_{N-1})^T\).

\partitem{a} State and prove the maximum principle for any grid function
\(V=\{V_j\}\) with values for \(j=0,1,\ldots,N\), that satisfies
\(\Delta_h V_j\ge0\) for \(j=1,2,\ldots,N-1\). Sketch a grid function for which
\(\Delta_h V_j\ge0\).

\partitem{b} Derive an expression for the local truncation error and find the
equation that relates the local trunction error and the global error
\(e_j=v(x_j)-U_j\).

\partitem{c} Use the maximum principle from part (a) to show that
\(\|e\|_\infty=O(h^2)\) as the space step \(h\to0\).

\partitem{d} Prove that the matrix \(A\) is nonsingular.